Robustness improvement of polyhedral...
Transcript of Robustness improvement of polyhedral...
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Robustness improvement of polyhedral meshmethod for airbag deployment simulations.TU Delft
Santiago Alagon Carrillo, Numerical Analysis
Daily Supervisor: Prof. dr. ir. C. Vuik,Company Supervisor: Ir. E.H. Tazammourti
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Outline
MADYMO and Gasflow2.
Cartesian Mesh and Cut-cells.
Discretization, Numerical Scheme and Coupling.
Exact Geometry.
Point-in-Polyhedron problem.
Search algorithm.
Topological tests.
Exact Geometry and Robustness issues.
Research questions.
Test Problems.
Looking for solutions and First results.
Future work.
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Madymo
MADYMO is a software package developed to analyze safetysystems in transport vehicles.
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GasflowGasflow is the module in charge of the numerical computations usedin MADYMO to simulate the deployment of airbags.
The solution is approximated using Euler’s equations
d
dt
∫Ω
qdΩ +
∫S
Φ(q, n)dS = S(q) (1)
where
q =
ρρuρvρwρe
, and Φ =
ρvn
ρvnu + p · nxρvnv + p · nyρvnw + p · nz
ρvnH
(2)
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Cartesian Mesh and Cut-cells 1.To discretize the domain a cut-cell Cartesian Mesh is used. Thediscretization is performed as follows:
I The whole computational domain is divided into cartesian,hexahedral, cells.
I The cells are flagged as active cells if they belong entirely tothe flow region, inactive if they belong entirely outside the flowregion and cut-cells if they intersect the boundary of the objectimmersed in the flow, the airbag.
I The geometry of the airbag is preserved by removing the inactiveregion of the cut-cells.
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Cartesian Mesh and Cut-cells 2.
I Only the active cells and the active regions of the cut-cells areused to compute the solution
Cartesian Mesh discretization is more suitable than body fitteddiscretization for this problem because
I The geometry changes at every time step. The cartesianmeshing process can be implemented as an automatic procedure.
I Preserves the geometry and results in conservative schemes.
I Cut-cells are de-coupled from the surface description.
I Refinement is possible to capture more detail of the geometry.
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Discretization, Numerical Scheme and Coupling.
I The boundary of the airbag is approximated by a Finite Elementtriangulation.
I Euler equation’s are discretized using a Finite Volume methodover hexahedral cells that cover the whole flow domain.
I The Finite Volume scheme is based on a Roe scheme usingflow-differencing splitting the approximate the flux derivatives.
I Boundary conditions are enforced through the specification ofthe fluxes at the boundary.
I At each time step one evaluation of FE and flow solver areperformed, loose coupling procedure.
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Exact Geometry
The exact geometry of the cut-cells and a robust method todetermine active and inactive cells as well as active and inactiveregions of each cut-cell are important to preserve accuracy andperformance of the finite volume solver.
Search algorithm Computes the exact geometry of the cut-cells and
classifies the cells and cut-cells regions into active and inactive.
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Search Algorithm 1
Search algorithmThe whole domain is decomposed into Blocks, which are sets of cells,each Block containing the same number of cells.Using the Block structure the search algorithm is performed into threestages.
I Global search level. This search determines a list of candidatetriangular segments, form the FE triangulation, that mayintersect each bock.
I Euler search. This search determines a set of triangularsegments that may intersect every cell.
I Exact Geometry. This search determines all intersectionsbetween Euler cells and FE-segments using the intersectioncandidate list. After that the face polygons, boundary faces andpolyhedrons of each cut cell are obtained.
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Search Algorithm 2
Figuur: (a) global search level, (b) Euler search level
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Search Algorithm 2
I The Global search and Euler search are based ontopological tests to determine the intersection of the cellsand FE segments.
I The Exact Geometry is obtained by constructing theintersection points of the FE triangulation edges and thecell faces.
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Topological tests 1Signed volume of a simplex
V (Tv0v1...vd ) =1
3!
va0 va1 va2 1vb0 vb1 vb2 1vc0 vc1 vc2 1vd0 vd1 vd2 1
(3)
Figuur: Signed Volume Property
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Topological tests 2
I Point test. For each FE node find in which cell it resides,see Figure(3a).
I Edge test. For each FE edge test wether it intersected by acertain cell’s face, see Figure(3b).
I Slice test. For each FE segment test wether it isintersected by a certain cell’s edge, see Figure(3c).
Figuur: Exact Tests
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Exact Geometry and Robustness issues 1State of the art in MADYMO:
I The implementation in MADYMO is capable of determining theexact geometry of the cut-cells.
I The implementation in MADYMO is capable of determining theinner and outer regions of the cut-cells in general cases, butthere are particular cases where it fails.
ImplementationThe airbag is a simple polyhedron. Considering a transversal cut theproblem can be equivalently stated in two-dimensions, this is the casefor each cell-face.
Figuur: Equivalence to two-dimensional problem
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Exact Geometry and Robustness issues 2
The outward normal to the FE triangulation is known, this determinesa consistent order for the points in the two-dimensional equivalentproblem.
Figuur: Consistent ordering
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Exact Geometry and Robustness issues 2
Determination of the inner and outer regions of the cut-cell’s facesusing the consistent ordering defined by the normal, the signedvolume of a simples and a vertex as a test point,
Figuur: Consistent ordering
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Exact Geometry and Robustness issues 3
Robustness problems arise when no vertex can be chosen as test point.
Figuur: No test point can be chosen systematically
.
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Research Question
How to determine which region in a cut-cell belongs to the flowand which out-side in a consistent way?In literature there are standard techniques to solve this so called Pointin polygon problem, such as the line crossing test
Figuur: Line crossing test
but this techniques have other problems such as
I Election of test point, round off error, use of geometric constructors,ignores information provided by normal vector.
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Test Problems1
Figuur: Test cases with no test point.
.
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Test Problems2
Figuur: Test cases with coplanar FE segments and cell faces.
.
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Looking for solutions and First results, 1.
Consistency in traversing direction 1.Two simple polygons sharing an edge traversed in acounter-clock wise direction result in the shared edge beingtraverses in opposite directions for each polygon.
Figuur: Simple polygons with coinciding edge.
.
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Looking for solutions and First results, 2.Consistency in traversing direction 2.
I For each cut-cell face, traverse each of the face polygons incounter-clock wise direction.
I For each of the polygons, if the counter-clock wise direction isnot consistent with the direction dictated by the normal of atleast one of the face segments then the face polygon isdetermined to be outside.
Figuur: Traversing directionconsistency.
.
I CCW Polygons:P1,P2,P3,P1 andP1,P3,P2,V3,V4,P1
I Face segments in normalinduced direction: P2,P3 andP3,P1
⇒ P1,P3,P2,V3,V4,P1 is outsideand P1,P2,P3,P1 is inside
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Looking for solutions and First results, 2.
This method has been implemented in Matlab for two of thetest cases. A correct Inside/Outside determination was obtainedfor both cases.
Figuur: First results
.
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Future work.
I Adapt and implement the method for test cases withco-planar triangular segments.
I Adapt and implement the method for non simple polygons.
I Explore other possible method based on a consistentelection of test points.
I Explore the use of ideas from the Level set Method